Artyom M. Grigoryan Brief Notes in Advanced DSP Fourier Analysis with MATLAB

Many interesting topics are studied in digital signal and image processing, and.
one of them is the theory and application of Fourier analysis. The Fourier.
transformation is the most used tool when analyzing and solving problems.
n the framework of linear systems that describe and approximate di?erent.
physical systems in practice. In digital signal processing (DSP), this transfor-.
mation gives the push for developing other fast discrete transformations, such.
as the Hadamard, cosine, and Hartley transformations. Another transforma-.
tion that is used in DSP, as well as in speech processing and communication, is.
the discrete Haar transformation. This is the ?rst orthogonal transformation.
developed after the Fourier transformation, which had been used as a basic.
stone to build the wavelet theory for the continuous-time signals. The Haar.
transformation used to be considered a transformation, that does not relate.
to the Fourier transformation. However, in the mathematical structure of the.
Fourier transformation, there is a unitary transformation, which is called the.
paired transformation and which coincides with the discrete Haar transfor-.
mation, up to a permutation. Such a similarity is only in the one-dimensional.
case; the discrete paired transformations exist in two- and multi-dimension.
cases as well, and they are not separable.
Discrete Fourier Transform.
nteger Fourier Transform.
Cosine Transform.
Hadamard Transform.
Paired Transform-Based Decomposition.
Fourier Transform and Multiresolution.